Binary-bed code written by Shawn Penson, Trent University, May 2002

For the understanding of this code, see the article by Slavin, Irvine and Penson, Int. J. of Mass and Heat Transfer, (paper II, submitted May 2002). This code calculates the thermal conductivity of a binary packed bed of ceramic spheroids when the large spheres have typically 10 times the diameter of the tiny spheres, so that the tiny spheres can be poured into a bed of the large ones to fill most of the interstices between the large spheres. The thermal conductivity of the small spheres alone is first calculated as described in the paper by Slavin, Arcas, Greenhalgh, Irvine and Marshall, Int. J. Mass and Heat Transfer, (paper I, in press May 2002) with the code given elsewhere at this web site, for a given pressure and for a range of temperatures (in kelvins). These temperatures are input into column 3, and the thermal conductivity for the small spheres into column 5. The calculated effective thermal conductivity of the binary bed is written into column 4. The individual conductances calculated are written into columns 10 to 18. These should be checked against the assumptions of the models as described in the papers. The subscripts t and l refer to the tiny and large spheroids, respectively. The other variables are defined in the references above.

The calculation below is for the binary-bed data of Enoeda et al. in Fusion Technol. 34 (1998) 877-881 for alumina in He gas, but some parameters for N2 are also included. Values appropriate to other data will need to be changed as should be obvious. Data that was not required for a specific calculation has been inactivated with a comment symbol (').

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'Enoeda data

T = col(3)

P=100 'pressure in kPa

Dl=2.99 'large particle diam in mm

Dt=0.34 'tiny particle diam in mm

R=Dl/2000 'large radius in microns

r=Dt/2000 'tiny radius in microns

N1= 10 'column to start recording the conductances

'FIXED CONSTANTS

pi=3.14

k=1.38e-23 'Boltzmann's constant

sig=5.67e-8 'Stefan-Boltzmann constant

L=sqrt(8/3)*R 'height of the close-packed cell

Pf= 0.658 'Enoeda 3mm 'experimental packing fraction for large particles

'Pf =0.589 'Enoeda 0.3mm

alphavp=(0.74/Pf)-1 '0.74 is the packing fraction for close-packed

Acp=2*sqrt(3)*R^2 'base area of the close-packed cell for large spheres

Av = Acp*alphavp 'base area of the "void" cell for large spheres

Nc =1.5 'no. of contacts per particle

thetamax = 60 'max. angle of integration for gas conduction near a contact

alphar = 1 'these four values of alpha are discussed in paper I

alphao=1

alphai=1

alphas = 2/1.5

hr = 0.19e-6 'sphere roughness Enoeda 3mm

'hr=0.26e-6 'sphere roughness Enoeda 1mm

'hr=0.26e-6 'sphere roughness Enoeda 0.3 mm guess since too small

'SOLID SPECIFICATIONS

'Ks=43.4-0.0656*T+2.93e-5*T^2 'thermal conductivity of the solid (alumina: see paper I)

eps=0.75 'thermal emissivity (depends on solid: see paper I)

a=1 'gas accommodation coefficient (see paper I)

'GAS SPECIFICATIONS

Cv=1.5*k 'gas heat capacity per molecule (He)

gamma=5/3 'Cp/Cv (He)

'gamma = 7/5 'Cp/Cv (N2)

n=P*1000/(k*T) 'gas number density

D=2.18e-10 'diameter of gas molecule (He)

'D=3.76e-10 'diameter of gas molecule (N2)

lambda=1/(sqrt(2)*n*pi*D*D) 'gas mean free path

v=sqrt(8*k*T/(pi*6.64e-27)) 'average velocity of a gas molecule (He)

'v=sqrt(8*k*T/(pi*4.68e-26)) 'average velocity of a gas molecule (N2)

j=1.8*lambda 'temperature jump distance (He)

'j=1.6*lambda 'temperature jump distance (N2)

'Gas conductivities at high pressure

Kg = 25*pi*Cv*v*lambda*n/64 'gas conductivity at high pressure (He)

'Kg= 0.777*25*pi*Cv*v*lambda*n/64 'gas conductivity at high pressure (N2) (see paper I)

'GEOMETRICAL PARAMETERS

thetat= arccos((R+hr-r)/R) 'polar angle from contact point of large spheres to centre of 'nearest tiny sphere

rt= sin(thetat)*R

rtt=rt 'radial distance to centre of first tiny sphere. Paper II, Fig. 2

thetatt=thetat 'angle to rtt: PaperII, Fig. 2

col(N1+8) = if(lambda*2/3 < 2*hr, 0,lambda*2/3-2*hr) 'value of r-lambda^2 (Paper II)

del=col(N1+8)

' if gas gap > 2/3 , mean free path is beyond the first tiny sphere, so

' integrate to first tiny sphere instead of to rlambda

thetalam=if(2*r > lambda*2/3,sqrt(del/R),thetatt) 'theta-lambda (Paper II, Fig. 1)

rlambda=if(2*r > lambda*2/3,sqrt(del*R),rtt) 'r-lambda (PaperII, Fig. 1)

'CALCULATION OF THE CONDUCTANCES

Gil=alphai*n*v*Cv*pi*rlambda^2*a*(gamma+1)/(16-(8*a)) 'Paper I, Eq. 8

col(N1+1)=Gil

Gol=alphao*Kg*pi*(1-exp(-R/lambda))*((cos(thetatt)-cos(thetalam))*R

+ (R+j + hr )*(ln(R+j+hr-R*cos(thetatt)) - ln(R+j+hr-R*cos(thetalam)))) 'Paper I, Eq. 9

col(N1+2)=Gol

w=r/2 'thickness of gas layer

'insert values of conductivity of tiny spheres in col 5

'see integral for Gt in Paper II, which gives

Gt=col(5)*pi*((cos(thetamax)-cos(thetatt))*R

+ (R+ hr )*(ln(R+hr-R*cos(thetamax)-w) - ln(R+hr-R*cos(thetatt)-w)))

'In the code below, the conductance of the two gas gaps of width w at the upper and lower large spheres (Paper II, Fig. 2) is calculated as that of a single gap of width 2w. These are the same except at low pressure where w~lambda.

' K2 = gas conductivity through gas gap. See Paper I, Eq.6.

K2=if(2*w > lambda*2/3,Kg,(a/(2-a))*((gamma+1)/8)*n*v*Cv*2*w)

Gg=K2*pi*R^2/(2*w+j)*(cos(thetatt)^2 - cos(thetamax)^2) 'conductance across gas gap 2w

Gtc=Gg*Gt/(Gg+Gt) ' total conductance through tiny sphere region

col(N1+3)=Gtc 'total conductance through the gas gaps + tiny spheres

col(N1+10)=Gt 'conductance through tiny spheres only

col(N1+11)=Gg 'conductance through one gas gap of width 2w

Gsl=alphas*Ks*pi*R^2/L 'conductance of one large sphere

col(N1)=Gsl

Grl=alphar*(pi*rtt^2)*4*sig*T^3/((2/eps)-1) 'conductance by rad'n between large spheres

col(N1+4)=Grl

Gtv=col(5)*Av/L 'conductance through the void region of the tiny spheres

col(N1+5) = Gtv

Gcontact=0 'assumes zero area of contact. See Paper I

col(N1+7)=Gcontact

'Gcp=conductance through the close packed regions of the large spheres. This assumes that all these conductances are in parallel with each other except for Gsl in series.

Gcp=Nc*( Gsl*(Gil+Gol+Gtc+Grl+Gcontact))/(Gsl+Gil+Gol+Gtc+Grl+Gcontact)

Geff = Gcp+Gtv 'effective conductance of the bed

Keff = (Geff)*(L/(Acp+Av)) 'effective thermal conductivity of the bed

col(4)=Keff